Lyapunov矩阵方程和Riccati矩阵方程解的一些估计
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摘要
Lyapunov矩阵方程和Riccati矩阵方程等线性和非线性矩阵方程足数值代数和非线性分析中研究和探讨的重要课题之一。它们在控制理论,运输理论,动态规划,梯形网络,统计过滤和统计学等科学和工程计算领域中有着广泛的应用。
本文采用控制不等式方法,给出了Lyapunov矩阵微分方程和连续代数Lyapunov矩阵方程解的特征值的和(包括迹)的上界;给出了两个矩阵乘积迹的上下界估计;使用相似变换,扩充了现有的一些基于Delta算子统一代数Lyapunov矩阵方程解的求解范围。
第一章介绍了Lyapunov矩阵方程和Riccati矩阵方程的应用背景和研究现状,给出本文所涉及的记号和引理。
第二章使用控制不等式方法,结合经典的特征值不等式和指数矩阵乘积的特征值不等式,给出了Lyapunov矩阵微分方程、连续代数Lyapunov矩阵方程解的特征值的和及其迹的上界。进一步,对Lyapunov矩阵微分方程使用特殊的相似变换,扩充了其求解范围,获得了具有更弱限制条件的Lyapunov矩阵微分方程、连续代数Lyapunov矩阵方程解的特征值的和及其迹的上界。
第三章利用矩阵的奇异值分解,结合控制不等式的技巧和经典的矩阵乘积迹的不等式,给出了两个矩阵乘积迹的上下界估计,并证明了所得结果改进了现有的一些结果,是现有的非对称情形中最好的估计。进一步,把迹界估计应用到连续代数Riccati矩阵方程中,利用H(o|¨)lder(Cauchy-Schwarts)不等式及特殊的凸函数不等式,给出了连续代数Riccati矩阵方程解的迹的上下界估计。
第四章利用可对角化矩阵的分解,结合控制不等式的方法和特殊的矩阵乘积迹的不等式,给出了一个是可对角化矩阵的两个矩阵乘积迹的上下界估计,证明了这个迹界估计比现有的一些结果更精确。进一步,利用所得结果,结合H(o|¨)lder(Cauchy-Schwarts)不等式及凸函数不等式,获得了连续代数Riccati矩阵方程解的迹的上下界估计,改进和推广了已有的一些结果。
第五章通过一些适当的变形,把基于Delta算子统一代数Lyapunov矩阵方程转换为连续代数的Lyapunov矩阵方程和离散代数的Lyapunov矩阵方程。使用特殊的相似变换,扩充了现有的一些基于Delta算子统一代数Lyapunov矩阵方程解的求解范围,结合控制不等式的技巧及其经典的矩阵积与和的特征值不等式,给出了这类矩阵方程解的上下界估计和解的数值界估计,改进了已有的结果。进一步,利用矩阵特征值和奇异值的特殊性质,给出了在一些特定要求下的某些特殊相似变换的相似矩阵的存在定理和相应算法。
Solving linear and nonlinear matrix equations such as the Lyapunov matrix equation and the Riccati matrix equation is one of important topics in the fields of numerical algebra and nonlinear analysis. Actually, they are widely used in areas of science and engineering computation, such as control theory, transport theory, dynamic programming, ladder networks, stochastic filtering and statistics.
In this paper, by using majorization inequalities, we obtain upper bounds on summations of eigenvalues (including the trace) of the solution for the Lyapunov matrix differential equation and the continuous algebraic Lyapunov matrix equation; and give new inequalities for the trace of the product of two matrices; by using similarity transformation, we expand the solution range of the existing unified algebraic Lyapunov equation based on Delta operator.
In chapter one, we present some background knowledge and recent works for the Lyapunov matrix equation and the Riccati matrix equation, and introduce some basic symbols and lemmas used in this paper.
In chapter two, we obtain upper bounds on summations of eigenvalues (including the trace) of the solution for the Lyapunov matrix differential equation and the continuous algebraic Lyapunov matrix equation by using majorization inequalities and classical eigenvalue inequalities with exponential matrix product eigenvalue inequalities. Further, by using similarity transformation, we expand the solution range of the existing the Lyapunov matrix differential equation and the continuous algebraic Lyapunov matrix equation, and obtain some new bounds with less constrictions of the solution for such a matrix equation.
In chapter three, by using matrix singularity decomposition, applying majorization inequalities and some classical matrix product trace inequalities, we propose new lower and upper trace bounds for the product of two matrices and prove them which improve some existing results are the tightest in nonsymmetric case. Further, by applying trace bounds in the continuous algebraic Riccati equation and using Holder (Cauchy-Schwarts) inequalities with special convex function inequalities, we obtain upper and lower bounds for the solution of the continuous algebraic Riccati equation.
In chapter four, we propose new lower and upper trace bounds for the product of two matrices in which one is diagonalizable by using diagonalizable matrix decomposition and applying majorization inequalities with some special matrix product trace inequalities, and prove them are more precise than some existing results. Further, by using our results in the continuous algebraic Riccati equation, applying Holder (Cauchy-Schwarts) inequalities and convex function inequalities, we obtain upper and lower bounds for the solution of the continuous algebraic Riccati equation which improve and extend some recent results.
In chapter five, by using some appropriate deformation, we change the unified algebraic Lyapunov equation based on Delta operator into the continuous algebraic Lyapunov equation and the discrete algebraic Lyapunov equation. We expand the solution range of the existing unified algebraic Lyapunov equation based on Delta operator by using similarity transformation, and obtain some new bounds of the solution for such a matrix equation by applying majorization inequalities with some classical matrix product with sum eigenvalue inequalities. Further, by using some special character of matrix eigenvalue and singular value, we give theorems and algorithm for the transformation matrix of some special similar transformation under certain special conditions.
本文采用控制不等式方法,给出了Lyapunov矩阵微分方程和连续代数Lyapunov矩阵方程解的特征值的和(包括迹)的上界;给出了两个矩阵乘积迹的上下界估计;使用相似变换,扩充了现有的一些基于Delta算子统一代数Lyapunov矩阵方程解的求解范围。
第一章介绍了Lyapunov矩阵方程和Riccati矩阵方程的应用背景和研究现状,给出本文所涉及的记号和引理。
第二章使用控制不等式方法,结合经典的特征值不等式和指数矩阵乘积的特征值不等式,给出了Lyapunov矩阵微分方程、连续代数Lyapunov矩阵方程解的特征值的和及其迹的上界。进一步,对Lyapunov矩阵微分方程使用特殊的相似变换,扩充了其求解范围,获得了具有更弱限制条件的Lyapunov矩阵微分方程、连续代数Lyapunov矩阵方程解的特征值的和及其迹的上界。
第三章利用矩阵的奇异值分解,结合控制不等式的技巧和经典的矩阵乘积迹的不等式,给出了两个矩阵乘积迹的上下界估计,并证明了所得结果改进了现有的一些结果,是现有的非对称情形中最好的估计。进一步,把迹界估计应用到连续代数Riccati矩阵方程中,利用H(o|¨)lder(Cauchy-Schwarts)不等式及特殊的凸函数不等式,给出了连续代数Riccati矩阵方程解的迹的上下界估计。
第四章利用可对角化矩阵的分解,结合控制不等式的方法和特殊的矩阵乘积迹的不等式,给出了一个是可对角化矩阵的两个矩阵乘积迹的上下界估计,证明了这个迹界估计比现有的一些结果更精确。进一步,利用所得结果,结合H(o|¨)lder(Cauchy-Schwarts)不等式及凸函数不等式,获得了连续代数Riccati矩阵方程解的迹的上下界估计,改进和推广了已有的一些结果。
第五章通过一些适当的变形,把基于Delta算子统一代数Lyapunov矩阵方程转换为连续代数的Lyapunov矩阵方程和离散代数的Lyapunov矩阵方程。使用特殊的相似变换,扩充了现有的一些基于Delta算子统一代数Lyapunov矩阵方程解的求解范围,结合控制不等式的技巧及其经典的矩阵积与和的特征值不等式,给出了这类矩阵方程解的上下界估计和解的数值界估计,改进了已有的结果。进一步,利用矩阵特征值和奇异值的特殊性质,给出了在一些特定要求下的某些特殊相似变换的相似矩阵的存在定理和相应算法。
Solving linear and nonlinear matrix equations such as the Lyapunov matrix equation and the Riccati matrix equation is one of important topics in the fields of numerical algebra and nonlinear analysis. Actually, they are widely used in areas of science and engineering computation, such as control theory, transport theory, dynamic programming, ladder networks, stochastic filtering and statistics.
In this paper, by using majorization inequalities, we obtain upper bounds on summations of eigenvalues (including the trace) of the solution for the Lyapunov matrix differential equation and the continuous algebraic Lyapunov matrix equation; and give new inequalities for the trace of the product of two matrices; by using similarity transformation, we expand the solution range of the existing unified algebraic Lyapunov equation based on Delta operator.
In chapter one, we present some background knowledge and recent works for the Lyapunov matrix equation and the Riccati matrix equation, and introduce some basic symbols and lemmas used in this paper.
In chapter two, we obtain upper bounds on summations of eigenvalues (including the trace) of the solution for the Lyapunov matrix differential equation and the continuous algebraic Lyapunov matrix equation by using majorization inequalities and classical eigenvalue inequalities with exponential matrix product eigenvalue inequalities. Further, by using similarity transformation, we expand the solution range of the existing the Lyapunov matrix differential equation and the continuous algebraic Lyapunov matrix equation, and obtain some new bounds with less constrictions of the solution for such a matrix equation.
In chapter three, by using matrix singularity decomposition, applying majorization inequalities and some classical matrix product trace inequalities, we propose new lower and upper trace bounds for the product of two matrices and prove them which improve some existing results are the tightest in nonsymmetric case. Further, by applying trace bounds in the continuous algebraic Riccati equation and using Holder (Cauchy-Schwarts) inequalities with special convex function inequalities, we obtain upper and lower bounds for the solution of the continuous algebraic Riccati equation.
In chapter four, we propose new lower and upper trace bounds for the product of two matrices in which one is diagonalizable by using diagonalizable matrix decomposition and applying majorization inequalities with some special matrix product trace inequalities, and prove them are more precise than some existing results. Further, by using our results in the continuous algebraic Riccati equation, applying Holder (Cauchy-Schwarts) inequalities and convex function inequalities, we obtain upper and lower bounds for the solution of the continuous algebraic Riccati equation which improve and extend some recent results.
In chapter five, by using some appropriate deformation, we change the unified algebraic Lyapunov equation based on Delta operator into the continuous algebraic Lyapunov equation and the discrete algebraic Lyapunov equation. We expand the solution range of the existing unified algebraic Lyapunov equation based on Delta operator by using similarity transformation, and obtain some new bounds of the solution for such a matrix equation by applying majorization inequalities with some classical matrix product with sum eigenvalue inequalities. Further, by using some special character of matrix eigenvalue and singular value, we give theorems and algorithm for the transformation matrix of some special similar transformation under certain special conditions.
引文
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[6]Pusz W,Woronowitz S L.Funcitional calculus for sequilinear forms and the purification mape[J].Rep.Math.Phys.,2003,1:459-463.
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[14] N. Komaroff. Diverse bounds for the eigenvalues of the continuous algebraic Riccati equation [J]. Ibid., 1994, 39:532-534.
[15] N. Komaroff. Upper bounds for the eigenvalues of the solution of the discrete Lyapunov matrix equation[J]. IEEE Trans. Automat. Contr., 1990, 35:468-469.
[16] Svetoslav. Savov and Ivan popchev. New Upper Estimates for the Solution of the Continuous Algebraic Lyapunov Equation [J]. IEEE Trans. Automat. Contr., 2004, 49:1841-1842.
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